Problem: In a spelling bee $50\%$ of the students were eliminated after the first round. Only $\frac{1}{3}$ of the remaining students were still in the contest after the second round. If 24 students were still in the contest after the second round, how many students began the contest?
Answer: The first round reduced the number of students by $\frac{1}{2}$ and the second round reduced the number of students by $\frac{1}{3}$.  After both reductions, $\frac{1}{2}\cdot\frac{1}{3}=\frac{1}{6}$ of the students who began the contest remain.  Therefore, $24\cdot6=\boxed{144}$ students began the contest.